The Evolution of Modern Metaphysics: Making Sense of Things (49 page)

Even so, what I have said so far does not convey the full force of the reason, nor, perhaps, its most important aspect.

Even within the shared commitment that analytic philosophers have to the study of language – whatever precise form that commitment takes – there is considerable latitude. In particular, what sort of language? Natural language? There is a broad division between those analytic philosophers who see themselves as dealing with language as it is and those who see themselves as dealing with language as it ought to be – but very often is not, absent the imposition of various kinds of reform on our ordinary ways of speaking. Frege himself certainly belonged to the second of these camps. He was largely contemptuous of natural language, which he held to suffer from all sorts of defects that hamper clear thinking. It was in this vein that he invented his own formal language, designed to enable him to address the questions in the philosophy of mathematics that particularly exercised him.
⁹

Were Frege’s attitude to be extended to the practice of metaphysics, this might appear to involve him in a clear and direct answer to the Novelty Question which I posed in §6 of the Introduction, the question whether there is scope for us, when we practise metaphysics, to make sense of things in ways that are radically new. It might appear that Frege would have been bound to say that there is, and bound indeed to say that we had better do so. In fact, however, there is something profoundly non-revisionary about Frege’s own use of his formal language. It was not intended to bring about radical changes in our sense-making. It was intended to exploit, nurture, and consolidate sense-making of ours that is already under way (cf.
Begriffsschrift
, Preface, pp. 6–7, and
Foundations
, §2). In saying this, I do not mean to suggest that Frege had no concern to bring about
any
changes
in connection with our sense-making. On the contrary, one thing that he wanted to do with his formal language, as part of the process of consolidation, was to bring about reform in those cases – those many cases – where, because of the imperfections of natural language, we merely
think
that we are making sense and we need the help of some such regimentation either to start making sense of the kind we think we are making or else to see that in fact there is no such sense to be made.
¹⁰
But when it came to introducing new concepts, Frege only ever showed an interest in drawing new boundaries in familiar regions of conceptual space, not in entering new regions; an interest, as we might say, in conceptual innovation but not in radical conceptual innovation (cf. §§64 and 88).
¹¹

None of this is enough to show that Frege would have given a conservative answer to the Novelty Question, had he addressed it. That he himself was not interested in radical changes in our sense-making does not mean that he would have denied the possibility of such a thing, either in metaphysics or in any other discipline. What primarily mattered for Frege was not whether we were making new sense or old sense, but simply whether we were making sense. And this at last brings us back to my reason for including him in my historical narrative. One of Frege’s greatest achievements was the way in which he made (linguistic) sense an object of philosophical scrutiny in its own right. Philosophers had certainly reflected on sense before. (See e.g.
Ch. 4
, §2, where we considered some of Hume’s ruminations on it.) There had even been attempts to subject different aspects of sense to close methodical investigation.
¹²
But these had been relatively piecemeal. Nobody previously, or at least nobody previously in the modern period,
had attempted to produce a
theory
of sense: a rigorous systematic comprehensive account of what sense is and how it functions. Frege did. I use the word ‘attempted’ advisedly. It is by no means uncontroversial that he succeeded. It is not even uncontroversial that what he attempted to do was something that could be done. We shall see scepticism of various kinds later in
Part Two
.
¹³
The fact remains that
Frege helped to provide a new focus in philosophy. Because he wanted to make sense of how we make mathematical sense, he was led to address some very general questions about sense itself. And, linguistic turn or no linguistic turn, he thereby helped to inaugurate a phase in my narrative in which due attention to sense came to be seen as an indispensable tool in the quest to
make
sense. Despite his lack of engagement with metaphysics, he is of immediate and obvious relevance to the story I have to tell.

2. The Project: Arithmetic as a Branch of Logic

Frege’s philosophical project is to show,
contra
Kant, that the truths of arithmetic are analytic. More specifically, it is to show that they are laws of logic (§87). ‘More specifically’, because a law of logic is a truth that is not only analytic but also composed (exclusively) of logical concepts. It may be analytic that all aunts are female, but it is not a law of logic. Neither the concept of an aunt nor the concept of being female is a logical concept.

Can Frege realize his project
just
by showing that the truths of arithmetic are composed of logical concepts? An analytic truth need not be composed of logical concepts; but is the converse perhaps true? Is a truth that is composed of logical concepts guaranteed to be analytic? No. A putative counterexample is that there are infinitely many non-logical objects
¹⁴
(if that is a truth; if it is not, then a putative counterexample is that there are only finitely many non-logical objects).
¹⁵

Frege’s task is twofold, then: to show that the truths of arithmetic are analytic, and to show that they are composed of logical concepts. It hardly appears that way to Frege, however. This is because, in Frege’s view, there is hardly anything, in the second case, to show. His notion of a logical concept is a concept that can be exercised in thought about any subject matter.
¹⁶
This is connected to the fact that he takes logical laws to govern thought (as such), not any of the more specific things that thought may be about (‘Thought’, p. 323/p. 58; cf. again §87, and cf. ‘Foundations of Geometry’, p. 338/pp. 427–428). And he takes it to be clear already that arithmetical concepts satisfy this condition; for
whatever
specific things we may think about, they can be counted (see e.g. ‘Formal Theories’, p. 112/pp. 94–95).
¹⁷

Insofar as it
is
clear already that arithmetical concepts are logical concepts, is it not likewise clear already that arithmetical truths are logical truths, that is laws of logic? If even Kantian things in themselves can be counted, for example, then must it not be the case that seven things in themselves, of some kind, and five things in themselves, of some disjoint kind, together constitute twelve things in themselves? Must not the truths of arithmetic extend as far as the concepts of arithmetic, which, if that is as far as coherent thought, straight away marks them out as laws of logic?

That is too quick. There is an equivocation here. ‘To extend as far as coherent thought’ may mean to extend to all that actually exists and can be an object of coherent thought, which is as much as is secured for the truths of arithmetic by the fact that its concepts are logical. Or it may mean to extend to all that can coherently be thought to exist, which is what is required of the truths of arithmetic for them to count as laws of logic. Even if the truths of arithmetic extend as far as coherent thought in the former, weaker sense, they may still depend, like the truth concerning how many non-logical objects there are, on some logical contingency about what actually exists. That is, they may fail to extend as far as coherent thought in the latter, stronger sense.

Even so, once we have got as far as agreeing that arithmetical concepts are logical concepts, which, however uncontentious it may seem to Frege,
already sets us apart from Kant
,
¹⁸
we have overcome what is probably the main obstacle to viewing the truths of arithmetic as laws of logic. And indeed there is a passage very early in the
Foundations
in which Frege all but gives the quick argument above for his thesis. Having indicated his agreement with Kant that the truths of geometry hold only of what can be given in spatial intuition,
¹⁹
he writes:

Conceptual thought alone can after a fashion shake off this yoke…. For purposes of conceptual thought we can always assume the contrary of some one or other of the geometrical axioms, without involving ourselves in any self-contradictions when we proceed to our deductions, despite the
conflict between our assumptions and our intuition. The fact that this is possible shows that the axioms of geometry are independent … of the primitive laws of logic, and consequently are synthetic. Can the same be said of the fundamental propositions of the science of number [i.e. arithmetic]? Here, we have only to try denying any one of them, and complete confusion ensues. Even to think at all seems no longer possible. The basis of arithmetic lies deeper, it seems, … than that of geometry. The truths of arithmetic govern all that is numerable. This is the widest domain of all; for to it belongs not only the actual, not only the intuitable, but everything thinkable. Should not the laws of number, then, be connected very intimately with the laws of thought [i.e. logical laws]? (§14)

It
looks
as if Frege is already where he wants to be. But no; he sees these considerations merely as lending plausibility to his thesis. He takes that thesis still to stand in need of proof.
²⁰
In particular, of course, he thinks he still needs to show that the truths of arithmetic are analytic. That is his great project.

Now although Frege intends nothing other by analyticity than what Kant intended (§3, n. 1), one of his chief services to philosophy is to provide a far clearer characterization of the notion than Kant ever did.
²¹
Frege’s characterization, in application to mathematics, is as follows:

When a proposition is called … analytic …, [this] is a judgement about the ultimate ground upon which rests the justification for holding it to be true.

This means that the question is … assigned, if the truth concerned is a mathematical one, to the sphere of mathematics. The problem becomes, in fact, that of finding the proof of the proposition, and of following it up right back to the primitive truths. If, in carrying out this process, we come only on general logical laws and on definitions, then the truth is an analytic one, bearing in mind that we must take account also of all propositions upon which the admissibility of any of the definitions depends. (§3)

To show that the truths of arithmetic are analytic, then, Frege needs to supply a set of suitable definitions and a set of ‘primitive’ logical laws, by which is presumably meant logical laws whose truth is beyond dispute and indeed – if questions are not to be begged – whose status as logical laws is beyond dispute, and then to demonstrate that the truths of arithmetic can be derived from the latter with the aid of the former. (This is reminiscent
of a procedure that Kant implicitly counted as sufficient for establishing that a truth is analytic, and that Leibniz explicitly counted as sufficient for establishing that a truth is a truth of reasoning: namely, to demonstrate that the denial of the truth can, by a finite process of analysis, be reduced to absurdity.
²²
) How then does Frege proceed?

3. The Execution of the Project

It is far beyond the scope of this chapter to supply a full answer to this question. But there are some features of Frege’s procedure that are especially worth noting in the context of our enquiry.

Despite his wariness of natural language, and despite his knowing full well that, in natural language, numerals sometimes have an adjectival use, Frege takes at face value their
nominal
use there, which is apparently to refer to particular objects: an example is the use of the numeral ‘four’ in the sentence, ‘The number of symphonies written by Schumann is the same as the number of gospels, namely four’ (§57). This certainly connects with the role played by numerals in arithmetic itself, where they likewise seem to function as names, used to refer to particular objects. The sentence ‘7 > 4’ has the same surface grammar as the sentence ‘Mount Everest is higher than Mount McKinley.’ Both sentences seem to relate one object to another. A crucial part of the project is therefore to say what exactly the objects referred to in arithmetic
are
– what numbers are (Introduction, pp. iff.).

Frege insists, again
contra
Kant, that they are not anything given in intuition (§12). But here already there is a complication. On Kant’s definition, intuition is simply ‘that through which [cognition] relates immediately to [objects]’ (Kant (
1998
), A19/B33). It is that whereby objects are immediately given to us. And Frege does not deny that numbers are given to us in some way. So does it not follow trivially that they are given to us in intuition?

In fact, granted what Kant goes on to say about intuition, and granted, for that matter, the reference to immediacy in his definition, it is clear that he means something that Frege is quite right to dissociate from his own conception of how numbers are given to us.
²³
What Kant means, as Frege
himself points out (§12), is a product of the faculty that he (Kant) calls sensibility. Sensibility is a faculty of pure passive receptivity. And it is to be contrasted with the faculty that Kant calls understanding, a faculty of spontaneity, whereby we actively think about what we passively receive. It is understanding which issues in concepts. (See
Ch. 5
, §4.)
²⁴
For Frege the practice of arithmetic does indeed require only something of the latter kind. Arithmetic is purely conceptual, in a sense in which even Kant would agree that logic is purely conceptual (Kant (
1998
), A52–53/B76–77).
²⁵
Otherwise, of course, its truths would be synthetic.